Abstract: If is real-valued, let and be the errors in best approximation to in the supremum norm by rational functions of type with real and complex coefficients, respectively. It has recently been observed that can occur for any , but for no is it known whether is zero or strictly positive. Here we show that both are possible: , but for . Related results are obtained for approximation on regions in the plane.

A. Ruttan, The length of the alternation set as a factor in determining when a best real rational approximation is also a best complex rational approximation, J. Approx. Theory 31 (1981), 230-243. MR 624011 (84d:41026)